Field control of quasiparticle decay in a quantum antiferromagnet

Dynamics in a quantum material is described by quantized collective motion: a quasiparticle. The single-quasiparticle description is useful for a basic understanding of the system, whereas a phenomenon beyond the simple description such as quasiparticle decay which affects the current carried by the quasiparticle is an intriguing topic. The instability of the quasiparticle is phenomenologically determined by the magnitude of the repulsive interaction between a single quasiparticle and the two-quasiparticle continuum. Although the phenomenon has been studied in several materials, thermodynamic tuning of the quasiparticle decay in a single material has not yet been investigated. Here we show, by using neutron scattering, magnetic field control of the magnon decay in a quantum antiferromagnet RbFeCl3, where the interaction between the magnon and continuum is tuned by the field. At low fields where the interaction is small, the single magnon decay process is observed. In contrast, at high fields where the interaction exceeds a critical magnitude, the magnon is pushed downwards in energy and its lifetime increases. Our study demonstrates that field control of quasiparticle decay is possible in the system where the two-quasiparticle continuum covers wide momentum-energy space, and the phenomenon of the magnon avoiding decay is ubiquitous.


I. EXPERIMENTAL DATA A. Integration ranges for inelastic neutron scattering spectra
The integration ranges for figures of inelastic neutron scattering spectra in this article are summarized in Table S1.For a spectrum sliced along an momentum-energy (q − ℏω) plane (false color map), the ranges perpendicular to q, two directions from 2a * − b * , b * , and c * , are displayed in reciprocal lattice unit (r.l.u.).For a constant q cut, the ranges at the q are shown.

TABLE S1.
Integration ranges for inelastic neutron scattering spectra.The definitions of the high symmetry positions in the momentum space is shown in Fig. 1b and 1c in the matin article.

B. Magnetic field dependence of constant q cut
Magnetic field dependence of the constant q cuts, the peak energies, and full width half maxima (FWHM L ) of the peaks at M point are shown in Fig. S1(a)-S1(c).The peak energies are reproduced using linear extended spin wave theory (LESW), as were those at K point in Fig. 3c in the main article.

A. Inelastic neutron scattering cross-section in RbFeCl 3
We employ the effective spin S = 1 Hamiltonian: where J c and J c2 are the nearest and next-nearest neighbor intrachain interactions, respectively; J ab is the nearest neighbor interaction in the ab-plane, constructing triangular lattice; D(>0) term is an easy-plane single-ion anisotropy, and the last term is the Zeeman term; the g value along the c direction is g c = 2.54 1 .The exchange constants are indicated in Fig. S2.The standard spin wave theory in linear approximation fails to reproduce the inelastic neutron scattering (INS) spectrum even at 0 T 2 .Hence, we used LESW, where the three eigenstates of the mean-field Hamiltonian were employed as the basis functions: ground state, the first excited state which is excited by the transverse component of the spin operator, T state, and the second excited state which is excited only by the longitudinal component, L state at 0 T. In contrast to the standard spin wave theory, employing the eigenstate (which has longitudinal fluctuation) captures the quantum effect of strong easyplane-type single-ion anisotropy and the hybridization between transverse and longitudinal fluctuation from a non-collinear magnetic structure [3][4][5][6][7] .In the present study we used the  recipe described in Ref. 3 for the calculation.At 0 T, the low energy mode ℏω 1 (q), highenergy mode ℏω 2 (q), and their relatives ℏω i (q + Q) and ℏω i (q − Q) (i =1, 2) were derived (six modes in total).Here Q is the propagation vector, Q = (1/3, 1/3, 0).Fitting the dispersion relation to the data at 0 T with the parameters of J c , J c2 , J ab , and D is depicted by the curves in Fig. 1(a) in the main article.Successful fit using the calculation means that the use of L state as one of the basis functions in LESW was good and a substantial amount of longitudinal fluctuation was present in the observed spectra of RbFeCl 3 .The parameters obtained were J c = −0.628(5)meV, J c2 = 0.104(5) meV, J ab = 0.048(1) meV, and D = 2.32(1) meV, which agrees with the previous study 8 .
The excitations in all fields appeared to have the inversion symmetry with respect to the Γ point as shown in Figs.2(a)-2(f) in the main article.This is explained by the superposition of excitations from two magnetic domains defined by the propagation vectors ±Q.Magnetic domains represented by +Q will hereafter be referred to as +QD, and −Q as −QD.The application of a magnetic field perpendicular to the spin plane of the 120 • structure induces a non-reciprocal magnon ε(q, +QD) ̸ = ε(−q, +QD) in each domain 3,9 .It was also found that the excitations from the two domains had the relation of ε(q, +QD) = ε(−q, −QD), and its superposition ε s (q) = ε(q, +QD) + ε(q, −QD) was reciprocal.The excitations from the two domains were observed to have equivalent scattered intensity, suggesting equivalent population for each domain.This result is similar to that reported in Ba 3 CoSb 2 O 9 10 .
The degeneracy of modes from the two magnetic domains +QD and −QD at the zero field was lifted by applying the magnetic field.Twelve modes are obtained: ℏω i (q, ±QD), ).Among these modes, the combination of ℏω i (q + Q, +QD) and ℏω i (q − Q, −QD), and ℏω i (q − Q, +QD) and ℏω i (q + Q, −QD) were related by inversion symmetry.The pairs of the combinations as modeled by LESW are exhibited in Figs.S3(a)-S3(x), with both dispersion and calculated cross sections.Owing to the strong single-ion anisotropy of the easy-plane type, the INS cross-section for ℏω i (q, ±QD) and S3(t)-S3(x) predicted scattered intensity for gapped and gapless branches in the low energy region.In total we have the four pairs of the gapped modes, ℏω 2 (q ± Q, ±QD), ℏω 2 (q ∓ Q, ±QD), ℏω 1 (q ± Q, ±QD), and ℏω 1 (q ∓ Q, ±QD), and two pairs of the gapless modes, ℏω 1 (q ± Q, ±QD) and ℏω 1 (q ∓ Q, ±QD).Sums of all 12 modes are presented in Extended spin wave theory treats longitudinal fluctuations within the linear approximation, leading to the successful semi-quantitative analysis on RbFeCl 3 .However, the anharmonic term is not fully considered via LESW, leading to an underestimate of J eff and an overestimate of D, where J eff = 2(−2J c1 + 2J c2 + 3J ab ).The observed ordered moment from the neutron diffraction experiment was reported to be 1.9 µ B at 1.45 K 11 , whereas the estimate from the mean-field solution in LESW was 0.94 µ B using g ab = 3.84 12 .The underestimate of the magnetic moment leads to the evaluation of the ground state of the system as being closer to the quantum critical point (QCP) than reality.This corresponds to the underestimate of J eff /D 13 .The underestimate of J was reported in the standard linear spin wave theory as well 14 .The saturation field of the bulk magnetization in the field applied along the c-axis was 14 T 15 , whereas the field estimated from LESW is 20 T. Because the saturation field reflects the magnitude of the easy-plane type anisotropy D, the overestimate of the field results in the overestimate of D. The true ground state of RbFeCl 3 is further from the QCP than estimated by the spin parameters from LESW; hence, the longitudinal component of the spin fluctuation would be overestimated.Please note that in the field control of magnon decay, the interaction between one-magnon and two-magnon continuum plays an important role, and the absolute values of the exchange constants are irrelevant.
The underestimate of J eff does not affect the discussion in the main article.

B. Cross section for longitudinal fluctuation
To illustrate the amount of longitudinal spin fluctuation, I XX /I calculated using LESW, where I XX is the cross section from the longitudinal spin correlation 3 and I is the total cross section, is shown for ℏω 1 (q±Q, ±QD) and ℏω 1 (q∓Q, ±QD) in Fig. S4 and ℏω 2 (q±Q, ±QD) and ℏω 2 (q ∓ Q, ±QD) in Fig. S5.The regions where the modes were not experimentally observed are hatched by gray.It was found that the ℏω 1 modes were dominated by the transverse fluctuation, whereas ℏω 2 modes had significant longitudinal fluctuation.

C. Two-magnon density of state
We calculated 2M-DoS, D(q, ω), by the sum of the 2M-DoS calculated in the local coordinate for each QD, D(q, ω, ±QD): where τ is a reciprocal vector and N is the number of spins.2M-DoS is shown in Figs.2(m)-2(r) in the main article.With the increase in magnetic field, the maximum energy increased, the peak sharpened, and the peak top increased.The onset of an additional peak appeared at 1.4 meV at 2 T.
The additional peak energy decreased and the peak became sharper with the field.refer to this simulation as the beam simulation.
A kernel containing the information about the sample shape, size, orientations along the axes, lattice constants, and UB matrix is then generated for the second part of the Monte Carlo simulation.MCViNE loads the beam simulation, and it evaluates the scattering with the sample rotating it for the same angular range used for the measurement.Finally, the scattered neutron trajectories are recorded on the detector array, reduced, and placed on a regular grid within the same momentum and energy transfer ranges used for the data.
The resolution ellipsoids in the 4-dimensional q-ℏω space are then simulated, fitted, and the resulting FWHMs saved in the final output file.The finer the grid, the longer these simulations will take.To achieve proper statistical average simulations have been performed with 10 million neutrons saved in the beam simulation.
The calculated one-magnon spectra, convoluted by the resolution ellipsoids, and integrated over the q range as indicated in Table S1 are shown in Figs.2g-2l in the main article.Constant q cuts of the calculated spectra, convoluted by the resolution function, and integrated over the q range indicated in Table S1 at Figure Spectrometer Type of spectrum q or q 2a * − b * b * c * Fig. 1a HRC Sliced spectrum K 1 -Γ -0.1 0.05 Fig. 1a HRC Sliced spectrum Γ -M 1 0.05 -0.05 Fig. 1a HRC Sliced spectrum M 1 -K 2 -0.05 0.05 Fig. 1a HRC Sliced spectrum K 2 -K 3 0.05 0.05 -Fig.2a-2f HYSPEC Sliced spectrum M 3 -M 4 -0.04 0.2 Fig. 3a HYSPEC Constant q cut (4/3, −2/3, 0) 0.04 0.04 0.2 Fig. 3b HYSPEC Constant q cut (1, 0, 0) 0.04 0.04 0.2 Fig. S1(a) HYSPEC Constant q cut (3/2, 0, 0) 0.04 0.04 0.2 Fig. S1 (a) Magnetic field dependences of constant q cuts at M (3/2, 0, 0).The peaks indicated by deep and light blue are ℏω 1 modes, and those by deep and light red are ℏω 2 modes.The horizontal bars represent the instrumental resolution.(b) Magnetic field dependence of peak energies.Red and blue solid curves are calculations of ℏω 1 and ℏω 2 using linear extended spin wave theory.See Supplementary Information II B and Fig. S3 for the details of the modes in the legends.(c) Magnetic field dependence of full width half maximum (FWHM L ).FWHM L = 0 means that the simple Gaussian function was employed for the fittings.Error bars for FWHM L = 0 represent instrumental resolution.
Fig.S2The crystal structure and exchange constants in RbFeCl 3 .

Fig. S3
Fig. S3 Magnetic field dependences of inelastic neutron scattering cross-section for the pair of mode.White and red curves represent dispersion relations for +QD and −QD of each pair, respectively, where QD is the magnetic domain represented by Q.
1,2) was negligible, and they are not shown.The cross-sections of the pairs for ℏω 2 in Figs.S3(b)-S3(f) and Figs.S3(h)-S3(l) were significant for only one of two branches in the high energy region at a given q.Conversely, those for ℏω 1 in Figs.S3(n)-S3(r)

Fig. S4
Fig. S4 Magnetic field dependences of I XX /I for ℏω 1 modes, where I XX is the cross section from the longitudinal spin correlation and I is the total cross section.Dashed line represents Brillouin zone edge.Circles, squares, and triangles represent high symmetry point of K, M, Γ points, respectively.q region where modes were not experimentally observed are hatched by gray.
Fig. S5 Magnetic field dependences of I XX /I for ℏω 2 modes, where I XX is the cross section from the longitudinal spin correlation and I is the total cross section.Dashed line represents Brillouin zone edge.Circles, squares, and triangles represent high symmetry point of K, M, Γ points, respectively.q region where the modes were not experimentally observed are hatched by gray.
Fig. S7 (a)-(c) The symbols indicate constant q cuts of the calculated spectra convoluted by the resolution function at K point in (a), Γ point in (b), and M point in (c).The solid curves are the fitting results by multiple Gaussians.The colored area indicates the component of the Gaussians, the central energy of which is the eigenenergy of the mode as defined in the legend.(d)-(f) ℏω dependence of FWHM of the Gaussian, denoted as FWHM G for each mode.